Talk:Parallel (geometry)

Parallel lines operator

Some info about the parallel lines operator?

A\|B

- Omegatron 04:24, August 9, 2005 (UTC)

Just found it in ISO 31-11 as Unicode character "∥" as well. --Abdull 22:09, 28 May 2006 (UTC)[reply]

parallel lines should never intersect

I think parallel lines should never intersect. This should make proving easier when you use Euclid's axiom 5 or Hilbert's axioms. i.e. if line m parallels to line l, then m!=l (this contradicts to the article in wiki). You can refer to http://mathworld.wolfram.com/Parallel.html Cheerful coffin 21:11, 18 October 2006 (UTC)[reply]

Just a suggestion

Shouldn't we give a precise definition for what a parallel line is in this article? For example, I propose the definition of: "a parallel line is any straight line that is relative to any other given straight line that has the same slope as the other and both of which do not poses the same exact line. In Euclidean geometry this means that the lines never intersect but this not necessarily the case in other non-flat geometries such as elliptic geometry or hyperbolic geometry. Also, by 'straight line' it should be concieved to mean the shortest distance between any two given points possible- in Euclidean this looks like the common straight line, but in other geometries this is a geodesic (exempli gratia: in spherical geometry, a sub-geometry of elliptic geometry, a straight line is really a geodesic following the curve of the sphere)."

Also, either on this article or that of elliptic (or both, I guess), I propose a more laymen's version of how to describe that parallel lines can intersect and how triangles can have more than a measurement of 180 degrees for their interior angles. I will do this now (in the elliptic geometry article) but if one wishes to, they can edit and remove it.

76.188.26.92 20:30, 31 May 2007 (UTC)[reply]

A line is not parallel with itself. "being parallel" is a textbook example of a relationship that is not reflexive. This page needs to be edited to have being parallel NOT include the degenerate case of l=m. —Preceding unsigned comment added by Jfrosen (talk) 05:57, July 27, 2007 (UTC)

This depends on the textbook & context used. For example, in high school I took geometry and "advanced math" (third-level algebra) in the same year. The former class said that two lines are parallel if they have no intersecting points. The latter class said that two lines are parallel if their slopes are identical. (Obviously these were limited to a coplaner space). The former definition mandates that a line cannot be parallel with itself; the latter does not. --Joe Sewell (talk) 19:07, 4 May 2011 (UTC)[reply]

Your welcome

I hope this article was useful:D —Preceding unsigned comment added by 216.78.42.3 (talk) 21:04, 2 April 2008 (UTC)[reply]

What do Parallel Lines look like in real life?

Parallel Lines are like railroad tracks that never intersect. —Preceding unsigned comment added by 63.215.28.94 (talk) 16:26, 6 April 2008 (UTC)[reply]

Only straight lines?

Do parallel lines imply that they must be perfectly straight lines? I mean aren't two curves that have equations y = sin(x) and y = sin(x) + 10 parallel to each other by the euclidean definition? - Temporary1139 (talk) 02:39, 19 May 2008 (UTC)[reply]

Parallel lines typically exhibit the following property: If you make a straight line from any point on line A to the nearest point on line B, then that straight line will intersect line A and line B perpendicularly. Your example does not meet that criterion. (OTOH, I just made that criterion up. I don't know if it is really a necessary property of parallel lines.) —Preceding unsigned comment added by 128.187.80.2 (talk) 23:37, 10 November 2008 (UTC)[reply]

Under the Old-School Euclidian Definition, Yes, they do. See: http://wiki.riteme.site/wiki/Euclidean_geometry http://wiki.riteme.site/wiki/Parallel_postulate However, as far as I understand, in the usual Cartesian space $\mathbb {R} ^{n}$ , no, they don't. Consider the vector definition of a line in parametric form. I don't have time to write it up, class soon, but if someone wants to take this be my guest. Basically as long as the distance between any two points $(x_{1},y_{1})$ and $(x_{2},y_{2})$ remains constant $\Rightarrow {\mathit {\mathbf {d} }}((x_{1},y_{1}),(x_{2},y_{2}))=(c_{1},c_{2}),(c_{1},c_{2})\in \mathbb {R} ^{+}\cup \{0\}$ , you're good.

In mathematics, a curve is not a line: a line has zero curvature within the space in which it is embedded, i.e. all lines are by definition perfectly straight*. If two curves such as sine waves do not meet this is trivial, for example many circles do not meet. If two curves remain a constant distance apart (as measured along any a line orthogonal to both curves) they are not parallels but isocurves, concentric circles are an example. Yes, a line between two nearest points on distinct parallel lines will be orthogonal. However, contrary to the previous reply, this applies especially in

\mathbb {R} ^{n}

which is Euclidean and even sometimes written as

\mathbb {E} ^{n}

. Parametric form has nothing to do with it.

* However if a line is drawn in a curved subspace, for example on a sphere

\mathbb {S} ^{2}

embedded in

\mathbb {E} ^{3}

, then although it is straight with respect to

\mathbb {S} ^{2}

it is not straight with respect to

\mathbb {E} ^{3}

, where we call it a great circle on the sphere.

HTH — Cheers, Steelpillow (Talk) 21:45, 23 March 2012 (UTC)[reply]

Parallel planes and hyper-planes?

This article lacks discussion of parallel planes and hyper-planes. It should discuss dihedral angles, and give methods for determining the angle between two hyper-planes. —Preceding unsigned comment added by 128.187.80.2 (talk) 21:53, 10 November 2008 (UTC)[reply]

home work

can parallel lines be curved —Preceding unsigned comment added by 86.179.85.25 (talk) 10:01, 16 January 2010 (UTC)[reply]

This is so stupid stupid stupid it doesn't make any sense that i typed in parallel lines and i got a movie this is some real bull... —Preceding unsigned comment added by 96.38.231.151 (talk) 18:24, 22 April 2010 (UTC)[reply]

Question About Formula (Legend)

Can you give the legend for the formula? for example what is b1 and b2? thank you —Preceding unsigned comment added by 203.201.160.66 (talk) 04:16, 28 December 2010 (UTC)[reply]

Symbol confusion in section

Hi all,

I think the italic l used in this section looks like a slash. Maybe it should be changed to the script ℓ to avoid confusion?

Thanks,

The Doctahedron, 03:02, 6 January 2012 (UTC)[reply]

I think this would cause larger problems with consistency than it would solve in readability. The standard in Mathematical literature is to use italics rather than script. You might try changing your browser settings so that your default font is serif rather than sans serif; sans serif is harder to read in general and this issue is one example.--RDBury (talk) 15:37, 6 January 2012 (UTC)[reply]

Comment at end of lead section

It says "In a non-Euclidean space, parallel lines are those that intersect only in the limit at infinity." This doesn't seem to make any sense. Can anyone confirm or clarify? Rschwieb (talk) 02:11, 22 November 2013 (UTC)[reply]

That is quite wrong, many spaces have no "infinity" and some of these are non-Euclidean (e.g. the surface of the sphere as a finite elliptic 2-space). I have deleted it, and also tried to make the lead more understandable to the absolute beginner. — Cheers, Steelpillow (Talk) 10:12, 22 November 2013 (UTC)[reply]

I thought so too: thank you! Rschwieb (talk) 15:11, 22 November 2013 (UTC)[reply]

About parallel Lines in a Non-Euclidean Space

I had added two lines about the property of "parallelism" in a curved surface (space). Those are removed and I was asked to get the source of them.

" We’ve known for a hundred years, since Albert Einstein, that space itself is fundamentally curved. We wrote a bit about this here. Lines and circles in real space simply do not do behave the way Euclid imagined, and so thinking in a non-Euclidean manner is fundamental to understanding the Universe. " --from http://www.qedcat.com/archive/97.html

On basis of many such reads, I added, "if a space is 180° twisted, any pair of parallel straight lines will intersect each other." I hope, Math Students will be able to understand this simple relativity.

Thanks. --Aaniya B (talk) 14:39, 16 January 2014 (UTC)[reply]

Hi, thank you for coming here to discuss it. I have looked at the paper you reference and I cannot find the idea you mention. Nor am I familiar with it from elsewhere, in fact quite the opposite. In particular, a smooth manifold has (by definition) no topological "point of twist". As a counter-example, a hyperbolic manifold is non-Euclidean and may be visibly twisted so it will not lie flat, yet can have many lines parallel to each other and which never meet. So I am reverting your edit again. If you still feel that it is valuable, please post it here for further discussion and do not restore it directly to the article again, as that might be seen as edit warring. — Cheers, Steelpillow (Talk) 17:33, 16 January 2014 (UTC)[reply]

Non-mathematical definition ?

This article is entirely mathematical definition based. May I suggest that the colloquial use of the word parallel is valid and should also be covered in the same article ?

For example: http://www.oxforddictionaries.com/us/definition/english/parallel

I think that most ordinary people (non-mathematically trained) would regard curved lines that remain equidistant from one another to be parallel. For example, railway lines are generally said to be parallel, but they are not necessarily straight. Lines of latitude are circles of different sizes, and are said to be parallels.

(Even the inner and outer handrails of a spiral staircase, which remain equidistant from one another in a horizontal (or spiral) plane, are parallel in my mind.) Darkman101 (talk) 17:02, 6 March 2016 (UTC)[reply]

The disambiguator (geometry) indicates that this is an article about the mathematical meaning of parallel. There is a disambiguation page for parallel that helps readers find other usages of the term, including several that you have mentioned and many that you haven't. Perhaps the hatnote should be expanded to point to that page (at present it only points to parallel line, a different disambiguation page). I would want to avoid the colloquial use of the term parallel in reference to equidistant curves on this page as it is more likely to confuse than clarify the concept for the natural set of readers for this article. Bill Cherowitzo (talk) 18:29, 6 March 2016 (UTC)[reply]

I did look at the disambiguation page for "parallel lines" but I saw nothing for "parallel". I agree that some kind of note should be added, at least to state that this narrow mathematical definition differs from the colloquial use of the term. As they are used to create parallel lines, do you think the parallel ruler and rolling ruler should also be on that page ? Darkman101 (talk) 02:02, 7 March 2016 (UTC)[reply]

(https://wiki.riteme.site/wiki/Parallel_rulers)(https://wiki.riteme.site/wiki/Rolling_ruler)

I'll take care of the hatnote. The section on constructions is a little thin and an important statement is buried in a footnote, so I think I will work on this section and can incorporate links to those technical tools when I do so. Bill Cherowitzo (talk) 05:24, 8 March 2016 (UTC)[reply]

Parallel lines redirect

I honestly feel like the redirect of Parallel lines should be removed. Why? There is a page with this exact title, (Blondie's 1978 album), with no parenthesis in it. If there should be a redirect with those words, it should be for that page. — Preceding unsigned comment added by The 10th Doctor (talk • contribs) 16:54, 12 July 2016 (UTC)[reply]

Something needs to be done. For now I have redirected Parallel lines to Parallel lines (disambiguation). At least the Blondie album can now be found. I am not sure of the official way to disambiguate this, but I believe that making Parallel lines into the disambig page would be a good option. Sorry, must dash now. — Cheers, Steelpillow (Talk) 17:11, 12 July 2016 (UTC)[reply]

image of "parallel lines and curves" confusing

It strikes me that to be consistent with the rest of the page the image of "parallel curves" should be removed. I actually looked at this page to figure out if it was acceptable to use the term parallel to describe curves that in 2d cartesian space for any x value, the y values of the two curves were a constant arithmetic difference from eachother. Seems like this is not "parallel" and I will stick to a wordier more precise definition. Anwyays, perhaps remove the image? Muniche (talk) 17:51, 15 November 2019 (UTC)[reply]

I don't think there's much danger, although bluelinking the terms in the caption to the diagram to the right articles might dispel that confusion before it arises. It's not as if anyone is claiming parallel curves are parallel lines. Illustrative pictures often contain things pertaining to related concepts outside the article. Rschwieb (talk)

Zero vector parallel to all other vectors?

I was searching for information to help answer a redditor on r/LinearAlgebra and it appears that the convention is that the zero vector is parallel to all other vectors because it has no direction. I could not find this information on Wikipedia and was hoping someone could find a source to verify this and add a brief mention at the beginning somewhere in or near the statement in the current article:

"Line segments and Euclidean vectors are parallel if they have the same direction or opposite direction (not necessarily the same length)." MHSchultzz (talk) 19:10, 28 November 2024 (UTC)[reply]

A more natural statement would be that the zero vector is orthogonal to all vectors since two vectors are orthogonal when their dot product is zero. — Rgdboer (talk) 01:06, 29 November 2024 (UTC)[reply]

I personally wouldn't consider the zero vector to have a direction at all, which for me would prevent it from being parallel to any other vector. This is a somewhat arbitrary matter of definitions in an edge case. Whether it's useful to consider the zero vector parallel to other vectors or not is going to depend on the context; most of the time it seems unlikely to be useful, but sometimes it might make sense. –jacobolus (t) 07:49, 29 November 2024 (UTC)[reply]